This is beyond a doubt the same result discussed here recently: ----- The chromatic number of the plane is at least 5. ----- In fact there's now a polymath project, led in part by Terry Tao, to try to simplify the proof or improve this bound. —Dan P.S. No offense, but this is a rather poor description of the problem. It doesn't even mention the plane! But then again, math is not The Grauniad's strong point: ----- The problem is as follows. Imagine a collection of dots connected by

lines. The dots can be arranged any way at all, the only rule is that all the connecting lines must be of equal length. For instance, in a square the diagonal would not be joined up, but the outer edges would be. Now, colour in all the dots so that no two connected points have the same colour. How many colours are required. For a square, the answer would be two. But the Hadwiger-Nelson problem asks what the minimum would be for any configuration -- even one that extends across a plane of infinite size./